Given any point $p \in \mathbb{R}^n$ and two matrices whose columns are orthonormal $X \in \mathbb{R}^{n \times j}$, $Y \in \mathbb{R}^{n \times (n - j)}$, such that $Y$ spans the orthogonal compliment of $X$.
Is it true that $$ \left| \left|\ pW \ \right| \right|_{1,1} \approx \left| \left|\ pX \ \right| \right|_{1,1} + \left| \left|\ pY \ \right| \right|_{1,1}? $$ where $W = [X | Y]$ is the concatenation of $X$ and $Y$ ($W \in \mathbb{R}^{n \times n}$).
And if it's true, how can we prove it? (It's known that the above is actually an equality when $L_{1,1}$ norm is replaced by the squared frobenius).
For more reference, do see the definition of $L_{1,1}$ Here.
I am asking this since if we compute it in MATLAB, then its not even equal (however for me I don't see why it wouldn't be equal)
Thanks in advance.